Probe outlined in my answer works in 3 nodes discovery when Istio presented. If livenessProbe is bad, than k8s will restart container even not allowing to start properly. I use internal Elastic ports (for node to node communication) to test liveness. These ports speak TCP.

      livenessProbe:
        tcpSocket:
          port: 9300
        initialDelaySeconds: 60 # it takes time from jvm process to start start up to point when discovery process starts
        timeoutSeconds: 10

          - name: discovery.zen.minimum_master_nodes
          value: "2"
          - name: discovery.zen.ping.unicast.hosts
          value: elastic

First Please check the logs using

kubectl logs <pod name> -n <namespacename>

You have to first run the init container and change the volume permissions.

you have to run the whole config as the user : 1000 also before the container of elasticsearch start you have to change the volume permission using init container.

apiVersion: apps/v1
kind: StatefulSet
metadata:
  labels:
    app : elasticsearch
    component: elasticsearch
    release: elasticsearch
  name: elasticsearch
spec:
  podManagementPolicy: Parallel
  replicas: 1
  revisionHistoryLimit: 10
  selector:
    matchLabels:
      app : elasticsearch
      component: elasticsearch
      release: elasticsearch
  serviceName: elasticsearch
  template:
    metadata:
      creationTimestamp: null
      labels:
        app : elasticsearch
        component: elasticsearch
        release: elasticsearch
    spec:
      containers:
      - env:
        - name: cluster.name
          value: <SET THIS>
        - name: discovery.type
          value: single-node
        - name: ES_JAVA_OPTS
          value: -Xms512m -Xmx512m
        - name: bootstrap.memory_lock
          value: "false"
        image: elasticsearch:6.5.0
        imagePullPolicy: IfNotPresent
        name: elasticsearch
        ports:
        - containerPort: 9200
          name: http
          protocol: TCP
        - containerPort: 9300
          name: transport
          protocol: TCP
        resources:
          limits:
            cpu: 250m
            memory: 1Gi
          requests:
            cpu: 150m
            memory: 512Mi
        securityContext:
          privileged: true
          runAsUser: 1000
        terminationMessagePath: /dev/termination-log
        terminationMessagePolicy: File
        volumeMounts:
        - mountPath: /usr/share/elasticsearch/data
          name: elasticsearch-data
      dnsPolicy: ClusterFirst
      initContainers:
      - command:
        - sh
        - -c
        - chown -R 1000:1000 /usr/share/elasticsearch/data
        - sysctl -w vm.max_map_count=262144
        - chmod 777 /usr/share/elasticsearch/data
        - chomod 777 /usr/share/elasticsearch/data/node
        - chmod g+rwx /usr/share/elasticsearch/data
        - chgrp 1000 /usr/share/elasticsearch/data
        image: busybox:1.29.2
        imagePullPolicy: IfNotPresent
        name: set-dir-owner
        resources: {}
        securityContext:
          privileged: true
        terminationMessagePath: /dev/termination-log
        terminationMessagePolicy: File
        volumeMounts:
        - mountPath: /usr/share/elasticsearch/data
          name: elasticsearch-data
      restartPolicy: Always
      schedulerName: default-scheduler
      securityContext: {}
      terminationGracePeriodSeconds: 10
  updateStrategy:
    type: OnDelete
  volumeClaimTemplates:
  - metadata:
      creationTimestamp: null
      name: elasticsearch-data
    spec:
      accessModes:
      - ReadWriteOnce
      resources:
        requests:
          storage: 10Gi

Check out the my yaml config and you can use. It's for single node of elasticsearch

The thing is that ElasticSearch itself has own health statuses (red, yellow, green) and you need to consider that in your configuration.

Here what I found in my own ES configuration, based on the official ES helm chart:

        readinessProbe:
          failureThreshold: 3
          initialDelaySeconds: 40
          periodSeconds: 10
          successThreshold: 3
          timeoutSeconds: 5

          exec:
            command:
              - sh
              - -c
              - |
                #!/usr/bin/env bash -e
                # If the node is starting up wait for the cluster to be green
                # Once it has started only check that the node itself is responding
                START_FILE=/tmp/.es_start_file

                http () {
                    local path="${1}"
                    if [ -n "${ELASTIC_USERNAME}" ] && [ -n "${ELASTIC_PASSWORD}" ]; then
                      BASIC_AUTH="-u ${ELASTIC_USERNAME}:${ELASTIC_PASSWORD}"
                    else
                      BASIC_AUTH=''
                    fi
                    curl -XGET -s -k --fail ${BASIC_AUTH} http://127.0.0.1:9200${path}
                }

                if [ -f "${START_FILE}" ]; then
                    echo 'Elasticsearch is already running, lets check the node is healthy'
                    http "/"
                else
                    echo 'Waiting for elasticsearch cluster to become green'
                    if http "/_cluster/health?wait_for_status=green&timeout=1s" ; then
                        touch ${START_FILE}
                        exit 0
                    else
                        echo 'Cluster is not yet green'
                        exit 1
                    fi
                fi